# Denumerable cellular families in $ZF$

### Kyriakos Keremedis

University of the Aegean, Karlovassi, Greece### Eliza Wajch

Siedlce University of Natural Sciences and Humanities, Poland

## Abstract

A denumerable cellular family of a topological space $X$ is a countably infinite collection of pairwise disjoint non-empty open sets of $X$. $IQDI$ is the sentence: For every infinite set $X$, the set of all finite subsets of $X$ has a countably infinite subset. Among other results, the following are proved in $ZF$: (i) $IQDI$ iff every infinite 0-dimensional Hausdorff space admits a denumerable cellular family; (ii) $IQDI$ implies that every infinite Hausdorff Baire space has a denumerable cellular family; (iii) if every metrizable Cantor cube is pseudocompact, then every non-empty countable collection of non-empty finite sets has a choice function; (iv) all metrizable Cantor cubes are paracompact; (v) the conjunction of $IQDI$ with a new modification of the Kinna–Wagner selection principle for families of finite sets implies that every infinite Boolean algebra has a tower and every infinite Hausdorff space has a denumerable cellular family.

## Cite this article

Kyriakos Keremedis, Eliza Wajch, Denumerable cellular families in $ZF$. Port. Math. 78 (2021), no. 3/4, pp. 281–321

DOI 10.4171/PM/2070